A joint statement revealed that Alfred McAlpine had granted Carillion a "limited period of access" to its books after receiving an increased cash-and-shares offer of £600M, or 585p a share.
Alfred McAlpine has previously rejected two approaches from Carillion – worth 560p and 570p a share respectively. It claimed these offers "materially undervalued" the company (NCE 18 October).
The move would be Carillion's second big acquisition in two years and follows its takeover of Mowlem for £313M last year. The deal would make it comfortably the UK's third biggest civil engineering contractor behind Balfour Beatty and Laing O'Rourke.
Carillion confirmed that its bid had the support of merchant bank Schroders and New Star Asset Management, which between them hold a combined 14.83% stake in the Alfred McAlpine business.
Carillion and Alfred McAlpine's joint statement said: "Subject to its detailed terms and conditions, the board of Alfred McAlpine has confirmed if such an offer were to be made it would receive its unanimous recommendation.
"Carillion and Alfred McAlpine believe that a combination of the two companies would present an excellent strategic fit, with the potential to create a company with a leading position in support services and integrated solutions."
But several analysts questioned the move and Carillion's share price fell on the news.
Alfred McAlpine and Carillion operate in similar markets and Alfred McAlpine this week also sold off the majority of its private finance portfolio of raods, schools and healthcare to Infrastructure Investors for £52.2M. It will retain its A13 PFI concession.